3.258 \(\int \frac{1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 \tan ^{-1}\left (\frac{2 \log (7 x)+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

(2*ArcTan[(1 + 2*Log[7*x])/Sqrt[3]])/Sqrt[3]

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Rubi [A]  time = 0.0180555, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{2 \log (7 x)+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(x + x*Log[7*x] + x*Log[7*x]^2)^(-1),x]

[Out]

(2*ArcTan[(1 + 2*Log[7*x])/Sqrt[3]])/Sqrt[3]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\log (7 x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \log (7 x)\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{1+2 \log (7 x)}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0285106, size = 22, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 \log (7 x)+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x + x*Log[7*x] + x*Log[7*x]^2)^(-1),x]

[Out]

(2*ArcTan[(1 + 2*Log[7*x])/Sqrt[3]])/Sqrt[3]

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Maple [A]  time = 0.007, size = 20, normalized size = 0.9 \begin{align*}{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,\ln \left ( 7\,x \right ) \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x+x*ln(7*x)+x*ln(7*x)^2),x)

[Out]

2/3*arctan(1/3*(1+2*ln(7*x))*3^(1/2))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+x*log(7*x)+x*log(7*x)^2),x, algorithm="maxima")

[Out]

integrate(1/(x*log(7*x)^2 + x*log(7*x) + x), x)

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Fricas [A]  time = 1.72742, size = 76, normalized size = 3.45 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \log \left (7 \, x\right ) + \frac{1}{3} \, \sqrt{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+x*log(7*x)+x*log(7*x)^2),x, algorithm="fricas")

[Out]

2/3*sqrt(3)*arctan(2/3*sqrt(3)*log(7*x) + 1/3*sqrt(3))

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Sympy [A]  time = 0.164534, size = 22, normalized size = 1. \begin{align*} \operatorname{RootSum}{\left (3 z^{2} + 1, \left ( i \mapsto i \log{\left (\frac{3 i}{2} + \log{\left (7 x \right )} + \frac{1}{2} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+x*ln(7*x)+x*ln(7*x)**2),x)

[Out]

RootSum(3*_z**2 + 1, Lambda(_i, _i*log(3*_i/2 + log(7*x) + 1/2)))

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Giac [A]  time = 1.79949, size = 26, normalized size = 1.18 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \log \left (7 \, x\right ) + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+x*log(7*x)+x*log(7*x)^2),x, algorithm="giac")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*log(7*x) + 1))